Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains

TL;DR

This paper establishes rigidity results for proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains by explicitly computing Bergman kernels, demonstrating the mappings' structural constraints in these complex, unbounded, and non-hyperbolic domains.

Contribution

It provides the first explicit formula for Bergman kernels on these domains and proves rigidity of proper holomorphic mappings, extending understanding of complex mappings in unbounded non-hyperbolic domains.

Findings

Rigidity results for proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains.

Explicit formulas for Bergman kernels on these domains.

Identification of unbounded weakly pseudoconvex domains with mapping rigidity.

Abstract

A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $\Sigma_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional generalized Fock-Bargmann-Hartogs domains. We therefore exhibit an example of unbounded weakly pseudoconvex domains on which the rigidity results of proper holomorphic mappings can be built.

Equations

$$D_{n,m}(\mu )=\{(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m:{\Vert w\Vert }^2<e^{-\mu {\Vert z\Vert }^2}\}\text{mbox}for\mu >0,$$

$$\begin{array}[]{l}\varphi_{U}:(z,w)\longmapsto(Uz,w),\quad U\in\mathcal{U}(n);\\ \\ \varphi_{U^{{}^{\prime}}}:(z,w)\longmapsto(z,U^{{}^{\prime}}w),\quad U^{{}^{\prime}}\in\mathcal{U}(m);\\ \\ \varphi_{v}:(z,w)\longmapsto(z+v,e^{-\mu\langle z,v\rangle-\frac{\mu}{2}{\left\lVert v\right\rVert}^{2}}w),\quad(v\in\mathbb{C}^{n}),\end{array}$$

$$\begin{array}{rl}\Sigma (\mathbf{n};\mathbf{p})& =\{({\zeta }_1,\cdots ,{\zeta }_r)\in {\mathbb{C}}^{n_1}\times \cdots \times {\mathbb{C}}^{n_r}:\sum _{k=1}^r\Vert {\zeta }_k{\Vert }^{2p_k}<1\},\end{array}$$

$$\phi ({\zeta }_1,\cdots ,{\zeta }_r)=({\gamma }_1({\zeta }_{\sigma (1)}),\cdots ,{\gamma }_r({\zeta }_{\sigma (r)}))$$

$${\phi }_a({\zeta }_1,{\zeta }_2,\cdots ,{\zeta }_r)=(T_a({\zeta }_1),{\zeta }_2({\psi }_a({\zeta }_1){)}^{1/2p_2},\cdots ,{\zeta }_r({\psi }_a({\zeta }_1){)}^{1/2p_r})$$

$${\psi }_a({\zeta }_1)=\frac{1-\Vert a{\Vert }^2}{(1-⟨{\zeta }_1,a⟩{)}^2}.$$

$$D_{n_0}^{\mathbf{n},\mathbf{p}}(\mu )=\{(z,w_{(1)},\cdots ,w_{(l)})\in {\mathbb{C}}^{n_0}\times {\mathbb{C}}^{n_1}\times \cdots \times {\mathbb{C}}^{n_l}:\sum _{j=1}^l{\Vert w_{(j)}\Vert }^{2p_j}<e^{-\mu {\Vert z\Vert }^2}\}(\mu >0),$$

$$\phi\circ f(z,w)=(z,w_{(\sigma(1))},\cdots,w_{(\sigma(l))})\left(\begin{array}[]{ccccc}A&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\Gamma_{1}&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\Gamma_{2}&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\ddots&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\Gamma_{l}\\ \end{array}\right),$$

$$f(z,w)=(z,w_{(\sigma(1))},\cdots,w_{(\sigma(l))})\left(\begin{array}[]{ccccc}A&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\Gamma_{1}&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\Gamma_{2}&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\ddots&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\Gamma_{l}\\ \end{array}\right),$$

$${\phi }_A:(z,w_{(1)},\cdots ,w_{(l)})⟼(zA,w_{(1)},\cdots ,w_{(l)});$$

$${\phi }_D:(z,w_{(1)},\cdots ,w_{(l)})⟼(z,(w_{(\sigma (1))},\cdots ,w_{(\sigma (l))})D);$$

$${\phi }_a:(z,w)⟼(z+a,w_{(1)}(e^{-2\mu <z,a>-\mu {\Vert a\Vert }^2}{)}^{\frac{1}{2p_1}},\cdots ,w_{(l)}(e^{-2\mu <z,a>-\mu {\Vert a\Vert }^2}{)}^{\frac{1}{2p_l}}),$$

$$D=\left(\begin{array}[]{cccc}\Gamma_{1}&\hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\Gamma_{2}&\hbox{}\hfil&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\ddots&\hbox{}\hfil\\ \hbox{}\hfil&\hbox{}\hfil&\hbox{}\hfil&\Gamma_{l}\\ \end{array}\right),$$

$$ϵ=\{\begin{array}{ll}1,& p_1=1\\ 0,& p_1\ne 1\end{array},\delta =\{\begin{array}{ll}1,& q_1=1\\ 0,& q_1\ne 1\end{array}.$$

$$F(z,w):(z,w_{(1)},\cdots ,w_{(l)})\to (z,w_{(1)}^2,w_{(2)},\cdots ,w_{(l)}).$$

$$\min \{n_{1+ϵ},n_2,\cdots ,n_l,n_1+\cdots +n_l\}\ge 2.$$

$$F:(z,w_{(1)})\to (\sqrt{2}z,w_{(1)}^2)$$

$$⟨f,g⟩={\int }_{\Omega }f(z)\overline{g(z)}dV(z)(f,g\in \mathcal{O}(\Omega )),$$

$$f(z)={\int }_{\Omega }f(w)K(z,w)dV(w)(z\in \Omega ).$$

$$⟨f,g⟩={\int }_{\Omega }f(z)\overline{g(z)}p(z)dV(z)(f,g\in \mathcal{O}(\Omega )).$$

$${\Omega }_{m,p}=\{(z,w)\in \Omega \times {\mathbb{C}}^m:\Vert w{\Vert }^2<p(z)\}.$$

$$K_m((z,w),(t,s))=\frac{m!}{{\pi }^m}\sum _{k=0}^{\infty }\frac{(m+1{)}_k}{k!}{K}_{A^2(\Omega ,p^{k+m})}(z,t)⟨w,s{⟩}^k,$$

$$K_{D_{n,m}}((z,w),(t,s))=\frac{m!{\mu }^n}{{\pi }^{m+n}}\sum _{k=0}^{\infty }\frac{(m+1{)}_k(k+m{)}^n}{k!}{e}^{\mu (k+m)⟨z,t⟩}⟨w,s{⟩}^k,$$

$$\alpha =({\alpha }_{(1)},\cdots ,{\alpha }_{(l)})\in ({\mathbb{R}}_{+}{)}^{n_1}\times \cdots \times ({\mathbb{R}}_{+}{)}^{n_l},$$

$$\beta (\alpha )=\frac{{\prod }_{i=1}^l\Gamma ({\alpha }_i)}{\Gamma (|\alpha |)};$$

$${\int }_{B_{+}^n}{r}^{2\alpha -1}d\text{V}(r)=\frac{\beta (\alpha )}{2^n|\alpha |},$$

$${\int }_{S_{+}^{n-1}}{w}^{2\alpha -1}d\sigma (w)=\frac{\beta (\alpha )}{2^{n-1}},$$

$${\int }_{\sum _{j=1}^l{\Vert w_{(j)}\Vert }^{2p_j}<t}{w}^{\alpha }{\bar{w}}^{\alpha }d\text{V}(w)=(\pi {)}^{n_1+\cdots +n_l}\frac{\prod _{i=1}^l\Gamma ({\alpha }_i+1)\prod _{i=1}^l\Gamma (\frac{|{\alpha }_{(i)}|+n_i}{p_i})}{\prod _{i=1}^l{p}_i\prod _{i=1}^l\Gamma (|{\alpha }_i|+n_i)\Gamma (\sum _{i=1}^l\frac{|{\alpha }_{(i)}|+n_i}{p_i}+1)}\cdot t^{\sum _{i=1}^l\frac{|{\alpha }_{(i)}|+n_i}{p_i}}$$

$${\int }_{\sum _{j=1}^l{\Vert w_{(j)}\Vert }^{2p_j}<t}{w}^{\alpha }{\bar{w}}^{\alpha }d\text{V}(w),$$

$$\text{eqref}eq2.5=(2\pi {)}^{n_1+\cdots +n_l}{\int }_{\genfrac{}{}{0ex}{}{\sum _{j=1}^l{\Vert s_{(j)}\Vert }^{2p_j}<t}{s_{ji}>0,1\le i\le n_j,1\le j\le l}}{s}^{2\alpha +1}d\text{V}(s).$$

Full text

Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains

Enchao Bi1**∗   &   Zhenhan Tu2

1School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, P.R. China

2School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, P.R. China

Email: bienchao@whu.edu.cn (E. Bi),  zhhtu.math@whu.edu.cn (Z. Tu)

††*∗*Corresponding author.

Abstract A generalized Fock-Bargmann-Hartogs domain $D_{n}^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in\mathbb{C}^{n}$ being a generalized complex ellipsoid $\Sigma_{z}({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional generalized Fock-Bargmann-Hartogs domains. We therefore exhibit an example of unbounded weakly pseudoconvex domains on which the rigidity results of proper holomorphic mappings can be built.

Key words: Automorphism groups, Bergman kernels, Generalized Fock-Bargmann-Hartogs domains, Proper holomorphic mappings

Mathematics Subject Classification (2010): 32A07·  32A25·  32H35·  32M05

1 Introduction

A holomorphic map $F:\Omega_{1}\rightarrow\Omega_{2}$ between two domains $\Omega_{1},\;\Omega_{2}$ in $\mathbb{C}^{n}$ is said to be proper if $F^{-1}(K)$ is compact in $\Omega_{1}$ for every compact subset $K\subset\Omega_{2}$. In particular, an automorphism $F:\Omega\rightarrow\Omega$ of a domain $\Omega$ in $\mathbb{C}^{n}$ is a proper holomorphic mapping of $\Omega$ into $\Omega$. There are many works about proper holomorphic mappings between various bounded domains with some requirements of the boundary (e.g., Bedford-Bell [3], Diederich-Fornaess [8], Dini-Primicerio [9] and Tu-Wang [24]). However, very little seems to be known about proper holomorphic mapping between the unbounded weakly pseudoconvex domains. There are also some works about automorphism groups of hyperbolic domains (e.g., Isaev [10], Isaev-Krantz [11] and Kim-Verdiani [14] ). In this paper, we mainly focus our attention on some unbounded non-hyperbolic weakly pseudoconvex domains.

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ is defined by

[TABLE]

where $\|\cdot\|$ is the standard Hermitian norm. The Fock-Bargmann-Hartogs domains $D_{n,m}(\mu)$ are strongly pseudoconvex domains in $\mathbb{C}^{n+m}$ with smooth real-analytic boundary. We note that each $D_{n,m}(\mu)$ contains $\{(z,0)\in\mathbb{C}^{n}\times\mathbb{C}^{m}\}\cong\mathbb{C}^{n}$. Thus each $D_{n,m}(\mu)$ is not hyperbolic in the sense of Kobayashi and $D_{n,m}(\mu)$ can not be biholomorphic to any bounded domain in $\mathbb{C}^{n+m}$. Therefore, each Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}.$

In 2013, Yamamori [25] gave an explicit formula for the Bergman kernels of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions. In 2014, by checking that the Bergman kernel ensures revised the Cartan’s theorem, Kim-Ninh-Yamamori [13] determined the automorphism group of the Fock-Bargmann-Hartogs domains as follows.

Theorem 1.1** (Kim-Ninh-Yamamori [13]).**

The automorphism group ${\rm Aut}(D_{n,m}(\mu))$ is exactly the group generated by all automorphisms of $D_{n,m}(\mu)$ as follows:

[TABLE]

where $\mathcal{U}(k)$ is the unitary group of degree $k,$ and $\langle\cdot,\cdot\rangle$ is the standard Hermitian inner product on $\mathbb{C}^{n}$.

Recently, Tu-Wang [23] has established the rigidity of the proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs domains as follows.

Theorem 1.2** (Tu-Wang [23]).**

If $D_{n,m}(\mu)$ and $D_{n^{\prime},m^{\prime}}(\mu^{\prime})$ are two equidimensional Fock-Bargmann-Hartogs domains with $m\geq 2$ and $f$ is a proper holomorphic mapping of $D_{n,m}(\mu)$ into $D_{n^{\prime},m^{\prime}}(\mu^{\prime})$, then $f$ is a biholomorphism between $D_{n,m}(\mu)$ and $D_{n^{\prime},m^{\prime}}(\mu^{\prime})$.

A generalized complex ellipsoid (also called generalized pseudoellipsoid) is a domain of the form

[TABLE]

where $\mathbf{n}=(n_{1},\cdots,n_{r})\in\mathbb{N}^{r}$ and $\mathbf{p}=(p_{1},\cdots,p_{r})\in(\mathbb{R}_{+})^{r}.$ In the special case where all the $p_{k}=1,$ the generalized complex ellipsoid $\Sigma(\mathbf{n};\mathbf{p})$ reduces to the unit ball in $\mathbb{C}^{n_{1}+\cdots+n_{r}}.$ Also, it is known that a generalized complex ellipsoid $\Sigma(\mathbf{n};\mathbf{p})$ is homogeneous if and only if $p_{k}=1$ for all $1\leq k\leq r$ (cf. Kodama [15]). In general, a generalized complex ellipsoid is not strongly pseudoconvex and its boundary is not smooth. The automorphism group ${\rm Aut}(\Sigma(\mathbf{n};\mathbf{p}))$ of $\Sigma(\mathbf{n};\mathbf{p})$ has been studied by Dini-Primicerio [9], Kodama [15] and Kodama-Krantz-Ma [16].

In 2013, Kodama [15] obtained the result as follows.

Theorem 1.3** (Kodama [15]).**

$(i)$* If $1$ does not appear in $p_{1},\cdots,p_{r}$, then any automorphism $\varphi\in{\rm Aut}(\Sigma(\mathbf{n};\mathbf{p}))$ is of the form*

[TABLE]

where $\sigma\in S_{r}$ is a permutation of the $r$ numbers $\{1,\cdots,r\}$ such that $n_{\sigma(i)}=n_{i},p_{\sigma(i)}=p_{i}$, $1\leq i\leq r$ and $\gamma_{1},\cdots,\gamma_{r}$ are unitary transformation of $\mathbb{C}^{n_{1}}(n_{\sigma(1)}=n_{1}),\cdots,\mathbb{C}^{n_{r}}(n_{\sigma(r)}=n_{r})$ respectively.

$(ii)$* If $1$ appears in $p_{1},\cdots,p_{r}$, we can assume, without loss of generality, that $p_{1}=1,p_{2}\neq 1,\cdots,p_{r}\neq 1$, then ${\rm Aut}(\Sigma(\mathbf{n};\mathbf{p}))$ is generated by elements of the form (1.1) and automorphisms of the form*

[TABLE]

where $T_{a}$ is an automorphism of the ball $\mathbb{B}^{n_{1}}$ in $\mathbb{C}^{n_{1}}$, which sends a point $a\in\mathbb{B}^{n_{1}}$ to the origin and

[TABLE]

In this paper, we define the generalized Fock-Bargmann-Hartogs domains $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ as follows:

[TABLE]

where $\mathbf{p}=(p_{1},\cdots,p_{l})\in(\mathbb{R}_{+})^{l},\;\mathbf{n}=(n_{1},\cdots,n_{l}),\;w_{(j)}=(w_{j1},\cdots,w_{jn_{j}})\in\mathbb{C}^{n_{j}},$ in which $n_{j}$ is a positive integer for $1\leq j\leq l.$ Here and henceforth, with no loss of generality, we always assume that $p_{i}\neq 1$ ($2\leq i\leq l$) for $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$.

Obviously, each generalized Fock-Bargmann-Hartogs domain $D_{n_{0}}^{\mathbf{n},\mathbf{p}}$ is an unbounded non-hyperbolic domain. In general, a generalized Fock-Bargmann-Hartogs domain is not a strongly pseudoconvex domain and its boundary is not smooth.

In this paper, we prove the following results.

Theorem 1.4**.**

Suppose $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ are two equidimensional generalized Fock-Bargmann-Hartogs domains. Let $f:\;D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ be a biholomorphic mapping. Then there exists $\phi\in\mathrm{Aut}(D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu))$ such that

[TABLE]

where $\sigma\in S_{l}$ is a permutation such that $n_{\sigma(j)}=m_{j}$, $p_{\sigma(j)}=q_{j}\;(1\leq j\leq l)$, $\sqrt{\frac{\nu}{\mu}}A\in\mathcal{U}(n)$ $(n:=n_{0}=m_{0})$, and $\Gamma_{i}\in\mathcal{U}(m_{i})\;(1\leq i\leq l)$.

Corollary 1.5**.**

Let $f:\;D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ be a biholomorphic mapping with $f(0)=0$. Then we have

[TABLE]

where $\sigma\in S_{l}$ is a permutation such that $n_{\sigma(j)}=n_{j}$, $p_{\sigma(j)}=p_{j}\;(1\leq j\leq l)$, $A\in\mathcal{U}(n_{0})$ and $\Gamma_{i}\in\mathcal{U}(n_{i})\;(1\leq i\leq l)$.

As a consequence, it is easy for us to prove the following results.

Theorem 1.6**.**

The automorphism group $\mathrm{Aut}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$ is generated by the following mappings:

[TABLE]

[TABLE]

[TABLE]

where $a\in\mathbb{C}^{n_{0}}$, $A\in\mathcal{U}(n_{0})$, $\sigma\in S_{l}$ is a permutation such that $n_{\sigma(j)}=n_{j}$, $p_{\sigma(j)}=p_{j}\;(1\leq j\leq l)$, and

[TABLE]

in which $\Gamma_{i}\in\mathcal{U}(n_{i})$ ($1\leq i\leq l$).

Now, for $\mathbf{p}$ and $\mathbf{q}$, we introduce notation:

[TABLE]

Theorem 1.7**.**

Suppose $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ are two equidimensional generalized Fock-Bargmann-Hartogs domains with $\min\{n_{1+\epsilon},n_{2},\cdots,n_{l},n_{1}+\cdots+n_{l}\}\geq 2$ and $\min\{m_{1+\delta},m_{2},\cdots,m_{l},m_{1}+\cdots+m_{l}\}\geq 2$. Then any proper holomorphic mapping between $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ must be a biholomorphism.

Remark 1.1**.**

The conditions $\min\{n_{1+\epsilon},n_{2},\cdots,n_{l}\}\geq 2$ can not be removed. For example, $n_{1}=1$ (i.e, $w_{1}\in\mathbb{C}$), $p_{1}\neq 1$, and

[TABLE]

Then $F$ is a proper holomorphic mapping between $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{n_{0}}^{\mathbf{n},\mathbf{q}}(\mu)$ where $\mathbf{q}=(p_{1}/2,p_{2},\cdots,p_{l})$. $F$ is not a biholomorphism.

Corollary 1.8**.**

Suppose $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is a generalized Fock-Bargmann-Hartogs domain with

[TABLE]

Then any proper holomorphic self-mapping of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ must be an automorphism.

Remark 1.2**.**

The conditions $n_{1}+\cdots+n_{l}\geq 2$ can not be removed. For instance, with no loss of generality, we can assume $n_{1}=1$ and $n_{i}=0$ ($2\leq i\leq l$). Then

[TABLE]

is a proper holomorphic self-mapping of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ which is not an automorphism.

The paper is organized as follows. In Section 2, using the explicit formula for the Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we prove that a proper holomorphic mapping between two equidimensional generalized Fock-Bargmann-Hartogs domains extends holomorphically to their closures and check that the Cartan’s theorem holds also for the generalized Fock-Bargmann-Hartogs domains. In Section 3, we exploit the boundary structure of generalized Fock-Bargmann-Hartogs domains to prove our results in this paper.

2 Preliminaries

2.1 The Bergman kernel of the domain $D_{n_{0}}^{\mathbf{n},\mathbf{p}}$

For a domain $\Omega$ in $\mathbb{C}^{n}$, let $A^{2}(\Omega)$ be the Hilbert space of square integrable holomorphic functions on $\Omega$ with the inner product:

[TABLE]

where $dV$ is the Euclidean volume form. The Bergman kernel $K(z,w)$ of $A^{2}(\Omega)$ is defined as the reproducing kernel of the Hilbert space $A^{2}(\Omega)$, that is, for all $f\in A^{2}(\Omega),$ we have

[TABLE]

For a positive continuous function $p$ on $\Omega$, let $A^{2}(\Omega,p)$ be the weighted Hilbert space of square integrable holomorphic functions with respect to the weight function $p$ with the inner product:

[TABLE]

Similarly, the weighted Bergman kernel $K_{A^{2}(\Omega,p)}$ of $A^{2}(\Omega,p)$ is defined as the reproducing kernel of the Hilbert space $A^{2}(\Omega,p)$. For a positive integer $m$, define the Hartogs domain $\Omega_{m,p}$ over $\Omega$ by

[TABLE]

Ligocka [17, 18] showed that the Bergman kernel of $\Omega_{m,p}$ can be expressed as infinite sum in terms of the weighted Bergman kernel of $A^{2}(\Omega,p^{k})\;(k=1,2,\cdots)$ as follows.

Theorem 2.1** (Ligocka [18]).**

Let $K_{m}$ be the Bergman kernel of $\Omega_{m,p}$ and let $K_{A^{2}(\Omega,p^{k})}$ be the weighted Bergman kernel of $A^{2}(\Omega,p^{k})$ $(k=1,2,\cdots)$. Then

[TABLE]

where $(a)_{k}$ denotes the Pochhammer symbol $(a)_{k}=a(a+1)\cdots(a+k-1)$.

The Fock-Bargmann space is the weighted Hilbert space $A^{2}(\mathbb{C}^{n},e^{-\mu\|z\|^{2}})$ on $\mathbb{C}^{n}$ with the Gaussian weight function $e^{-\mu\|z\|^{2}}\;(\mu>0)$. The reproducing kernel of $A^{2}(\mathbb{C}^{n},e^{-\mu\|z\|^{2}})$, called the Fock-Bargmann kernel, is ${\mu^{n}e^{\mu\langle z,t\rangle}}/{\pi^{n}}$ (see Bargmann [2]). Thus, the Fock-Bargmann-Hartogs domain $D_{n,m}=\{(z,w)\in\mathbb{C}^{n}\times\mathbb{C}^{m}:\left\lVert w\right\rVert^{2}<e^{-\mu{\left\lVert z\right\rVert}^{2}}\}\;(\mu>0)$ and the Fock-Bargmann space $A^{2}(\mathbb{C}^{n},e^{-\mu\|z\|^{2}})$ are closely related. In 2013, using Theorem 2.1 and the expression of the Fock-Bargmann kernel, Yamamori [25] gave the Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$ as follows.

Theorem 2.2** (Yamamori [25]).**

The Bergman kernel of the Fock-Bargmann-Hartogs domain $D_{n,m}$ is given by

[TABLE]

where $(a)_{k}$ denotes the Pochhammer symbol $(a)_{k}=a(a+1)\cdots(a+k-1)$.

Following the idea of Theorem 2.1, we compute the Bergman kernel for the generalized Fock-Bargmann-Hartogs domain $D_{n_{0}}^{\mathbf{n},\mathbf{p}}$. In order to compute the Bergman kernel, we first introduce some notation.

Let

[TABLE]

where $\alpha_{(i)}=(\alpha_{i1},\cdots,\alpha_{in_{i}})\in(\mathbb{R}_{+})^{n_{i}}$ for $1\leq i\leq l.$ For $\alpha\in(\mathbb{R}_{+})^{n}$, we define

[TABLE]

see D’Angelo [7]. Here $\Gamma$ is the usual Euler Gamma function.

Lemma 2.3** (D’Angelo [7], Lemma 1).**

Suppose $\alpha\in(\mathbb{R}_{+})^{n}$. Then we have

[TABLE]

[TABLE]

where $d\textrm{V}$ is the Euclidean $n$-dimensional volume form, $d\textrm{S}$ is the Euclidean $(n-1)$-dimensional volume form, and the subscript $``+"$ denotes that all the variables are positive, that is, $B_{+}^{n}=B^{n}\cap(\mathbb{R}_{+})^{n}$ and $S_{+}^{n-1}=S^{n-1}\cap(\mathbb{R}_{+})^{n}$, in which $B^{n}$ is the unit ball in $\mathbb{R}^{n}$ and $S^{n-1}$ is the unit sphere in $\mathbb{R}^{n}$.

Theorem 2.4**.**

Suppose $\alpha=(\alpha_{(1)},\cdots,\alpha_{(l)})\in(\mathbb{R}_{+})^{n_{1}}\times\cdots(\mathbb{R}_{+})^{n_{l}},\;\alpha_{(i)}=(\alpha_{i1},\cdots,\alpha_{in_{i}})\in(\mathbb{R}_{+})^{n_{i}},1\leq i\leq l.$ Then we have the formula:

[TABLE]

Proof.

For the integral

[TABLE]

by applying the polar coordinates $w=se^{i\theta}$ (namely, $w_{ij}=s_{ij}e^{i\theta_{ij}},$ $1\leq j\leq n_{i},\;1\leq i\leq l$, $s=(s_{(1)},\cdots,s_{(l)})$), we have

[TABLE]

Using the spherical coordinates in the variables $s_{(1)},s_{(2)},\cdots,s_{(l)}$ respectively, we get

[TABLE]

Let ${\rho_{i}}^{p_{i}}=r_{i},1\leq i\leq l$. Then we have $d\rho_{i}=\frac{1}{p_{i}}\rho_{i}^{1-p_{i}}dr_{i}=\frac{1}{p_{i}}r_{i}^{\frac{1}{p_{i}}-1}dr_{i}$. Therefore, Lemma 2.3 and the above formulas yield

[TABLE]

Let $r=(r_{1},r_{2},\cdots,r_{l})\in(\mathbb{R}_{+})^{l}$ and $k:=t^{-\frac{1}{2}}r$. Then $dr=t^{\frac{l}{2}}dk$. After a straightforward computation, we obtain that (2.2) equals

[TABLE]

Applying Lemma 2.3 to the above formula, we get

[TABLE]

where $\alpha^{{}^{\prime}}=(\frac{|\alpha_{(1)}|+n_{1}}{p_{1}},\cdots,\frac{|\alpha_{(l)}|+n_{l}}{p_{l}})\in(\mathbb{R}_{+})^{l}$. ∎

Now we consider the Hilbert space $A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$ of square-integrable holomorphic functions on $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$.

Lemma 2.5**.**

*Let $f\in A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$. Then

[TABLE]

*where the series is uniformly convergent on compact subsets of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu),$ $f_{\alpha}(z)\in A^{2}(\mathbb{C}^{n_{0}},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}})$ for any $\alpha=(\alpha_{(1)},\cdots,\alpha_{(l)})\in\mathbb{N}^{n_{1}}\times\cdots\times\mathbb{N}^{n_{l}},\;\alpha_{(i)}=(\alpha_{i1},\cdots,\alpha_{in_{i}})\in\mathbb{N}^{n_{i}},1\leq i\leq l$, $\lambda_{\alpha}=\sum\limits_{i=1}^{l}\frac{|\alpha_{(i)}|+n_{i}}{p_{i}}$, in which $A^{2}(\mathbb{C}^{n},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}})$ denotes the space of square-integrable holomorphic functions on $\mathbb{C}^{n}$ with respect to the measure $e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}}\textrm{d}V_{2n}$.

Proof.

Since $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is a complete Reinhardt domain, each holomorphic function on $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is the sum of a locally uniformly convergent power series. Thus, for $f\in A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)),$ we have

[TABLE]

where the series is uniformly convergent on compact subsets of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu).$ We choose a sequence of compact subsets $D_{k}\;(1\leq k<\infty)$

[TABLE]

where ${B(0,k)}$ is the ball in $\mathbb{C}^{n_{0}+n_{1}+\cdots+n_{l}}$ of the radius $k$. Obviously, $D_{k}\Subset D_{k+1}$ and $\bigcup\limits_{k=1}^{\infty}D_{k}=D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. Since $D_{k}$ is a circular domain, then

[TABLE]

in the Hilbert space $A^{2}(D_{k})$. Hence we have

[TABLE]

Since $f(z,w)\in A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$, we have

[TABLE]

Then $f_{\alpha}(z)w^{\alpha}\in A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$. Therefore,

[TABLE]

[TABLE]

By (2.1), it follows

[TABLE]

Consequently, $f_{\alpha}(z)\in A^{2}(\mathbb{C}^{n_{0}},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}}),$ where $\lambda_{\alpha}=\sum\limits_{i=1}^{l}\frac{|\alpha_{(i)}|+n_{i}}{p_{i}}$. ∎

Lemma 2.5 implies that $f(z)w^{\alpha}$ where $f(z)\in A^{2}(\mathbb{C}^{n_{0}},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}})$ form a linearly dense subset of $A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$. Now we can express the Bergman kernel of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ as follows.

Theorem 2.6**.**

The Bergman kernel of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ can be expressed by the following form

[TABLE]

where $\alpha=(\alpha_{(1)},\cdots,\alpha_{(l)})\in\mathbb{N}^{n_{1}}\times\cdots\times\mathbb{N}^{n_{l}},\;\alpha_{(i)}=(\alpha_{i1},\cdots,\alpha_{in_{i}})\in\mathbb{N}^{n_{i}},1\leq i\leq l$, and

[TABLE]

Proof.

Since $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is a complete Reinhardt domain, it follows

[TABLE]

where the sum is locally uniformly convergent, by the invariance of the Bergman kernel $K_{D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)}$ on $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ under the unitary subgroup action

[TABLE]

For any $\alpha=(\alpha_{(1)},\cdots,\alpha_{(l)})\in\mathbb{N}^{n_{1}}\times\cdots\times\mathbb{N}^{n_{l}}$ with $\alpha_{(i)}=(\alpha_{i1},\cdots,\alpha_{in_{i}})\in\mathbb{N}^{n_{i}}\;(1\leq i\leq l)$, any $f(z)\in A^{2}(\mathbb{C}^{n_{0}},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}})\;(\lambda_{\alpha}=\sum\limits_{i=1}^{l}\frac{|\alpha_{(i)}|+n_{i}}{p_{i}})$, we have $f(z)w^{\alpha}\in A^{2}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$. Thus

[TABLE]

By Bargmann [2], we get that the Bergman kernel of $A^{2}(\mathbb{C}^{n_{0}},e^{-\mu\lambda_{\alpha}{\left\lVert z\right\rVert}^{2}})$ can be described by the form

[TABLE]

Thus we obtain

[TABLE]

This completes the proof. ∎

The transformation rule for Bergman kernels under proper holomorphic mapping (e.g., Th. 1 in Bell [4]) is also valid for unbounded domains (e.g., see Cor. 1 in Trybula [21]). Note that the coordinate functions play a key role in the approach of Bell [4] to extend proper holomorphic mapping, but, in general, are no longer square integrable on unbounded domains. In order to overcome the difficulty, by combining the transformation rule for Bergman kernels under proper holomorphic mapping in Bell [4] and our explicit form (2.4) of the Bergman kernel function for $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$, we prove that a proper holomorphic mapping between two equidi- mensional generalized Fock-Bargmann-Hartogs domains extends holomorphically to their closures as follows.

Lemma 2.7**.**

Suppose that $f:\;D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ is a proper holomorphic mapping between two equidimensional generalized Fock-Bargmann-Hartogs domains. Then $f$ extends holomorphically to a neighborhood of the closure $\overline{D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)}$.

In fact, using the explicit form (2.4) of the Bergman kernel function for $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$, we immediately have Lemma 2.7 by a slightly modifying the proof of Th. 2.5 in Tu-Wang [23].

2.2 Cartan’s Theorem on the $D_{n_{0}}^{\mathbf{n},\mathbf{p}}$

Suppose $D$ is a domain in $\mathbb{C}^{N}$ and let $K_{D}(z,w)$ be its Bergman kernel. From Ishi-Kai [12], we know that if the following conditions are satisfied:

$(a)\quad K_{D}(0,0)>0$;

$(b)\quad T_{D}(0,0)\textrm{ is positive definite},$

where $T_{D}$ is an $N\times N$ matrix

[TABLE]

Then the Cartan’s theorem can also be applied to the case of unbounded circular domains. The above conditions are obviously satisfied by the bounded domain.

Kim-Ninh-Yamamori [13] proved the following result.

Lemma 2.8** (Kim-Ninh-Yamamori [13], Th. 4).**

Suppose that $D$ is a circular domain and its Bergman kernel satisfies the above conditions $(a)$ and $(b)$. If $\varphi\;(\in\mathrm{Aut}(D))$ preserves the origin, then $\varphi$ is a linear mapping.

Ishi-Kai [12] proved the generalization of Lemma 2.8 as follows.

Lemma 2.9** (Ishi-Kai [12], Prop. 2.1).**

Let $D_{k}$ be a circular domain (not necessarily bounded) in $\mathbb{C}^{N}$ with $0\in D_{k}\;(k=1,2),$ and let $\varphi:D_{1}\rightarrow D_{2}$ be a biholomorphism with $\varphi(0)=0$. If $K_{D_{k}}(0,0)>0$ and $T_{D_{k}}(0,0)$ is positive definite $(k=1,2)$, then $\varphi$ is linear.

Therefore, by using the expressions of Bergman kernels of generalized Fock-Bargmann-Hartogs domains, we have the following result.

Theorem 2.10**.**

Suppose that $\varphi:D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ be a biholomorphic mapping between two equidimensional generalized Fock-Bargmann-Hartogs domains with $\varphi(0)=0$. Then $\varphi$ is linear.

Proof.

By using the expressions (2.4) of Bergman kernels of generalized Fock-Bargmann-Hartogs domains and a straightforward computation, we show that the Bergman kernel of every generalized Fock-Bargmann-Hartogs domain satisfies the above conditions (a) and (b). So we get Th. 2.10 by Lemma 2.9. ∎

3 Proof Of The Main Theorem

To begin, we exploit the boundary structure of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ which is comprised of

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

Now we give the following proposition.

Proposition 3.1**.**

$(1)$* The boundary $b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is a real analytic hypersurface in $\mathbb{C}^{n_{0}+n_{1}+\cdots+n_{l}}$ and $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is strongly pseudoconvex at all points of $b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$.

$(2)$ $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is weakly pseudoconvex but not strongly pseudoconvex at any point of $b_{1}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and is not smooth at any point of $b_{2}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$.*

Proof.

Let

[TABLE]

Then $\rho$ is a real analytic definition function of $b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. Fix a point $(z_{0},w_{(1)0},\cdots,w_{(l)0})\in b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and let $T=(\zeta,\eta_{(1)},\cdots,\eta_{(l)})\in T_{(z_{0},w_{(1)0},\cdots,w_{(l)0})}^{1,0}(b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)).$ Then by definition, we know that

[TABLE]

[TABLE]

[TABLE]

Thanks to (3.1), (3.2) and (3.3), the Levi form of $\rho$ at the point $(z_{{0}},w_{(1)0},\cdots,w_{(l)0})$ can be computed as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

by the Cauchy-Schwarz inequality, for all $T=(\zeta,\eta_{(1)},\cdots,\eta_{(l)})\in T_{(z_{0},w_{(1)0},\cdots,w_{(l)0})}^{1,0}(b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$. Obviously, if $\zeta\neq 0$, then $L_{\rho}(T,T)>0$.

On the other hand, combining with (3.1), (3.2) and (3.3), we know that the equality holds if and only if

[TABLE]

[TABLE]

[TABLE]

Suppose $\zeta=0$, then $T=(\zeta,\eta_{(1)},\cdots,\eta_{(l)})\neq 0$ implies that there exists $\eta_{i_{0}}\neq 0$. If $L_{\rho}(T,T)=0$ for all $T\neq 0\in T_{(z_{0},w_{(1)0},\cdots,w_{(l)0})}^{1,0}(b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$, then by (3.1), (3.2), (3.3) and (3.6), we have $\eta_{k}=0$ ($1\leq k\leq l$). This is a contradiction.

When there exists $j_{0}\geq 1+\epsilon$ such that ${\left\lVert w_{(j_{0})0}\right\rVert}^{2}=0$ and $p_{j_{0}}>1$, then $(z_{0},w_{(1)0},\cdots,w_{(l)0})\in b_{1}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. Let $T_{0}=(0,\cdots,\eta_{(j_{0})},0,\cdots,0),\;\|\eta_{(j_{0})}\|\neq 0$. Then $L_{\rho}(T_{0},T_{0})=0$. Hence $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is weakly pseudoconvex but not strongly pseudoconvex on any point of $b_{1}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$.

It is obvious that $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is not smooth at any point of $b_{2}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. The proof is completed. ∎

Lemma 3.1** (Tu-Wang [24]).**

Let $\Sigma(\mathbf{n};\mathbf{p})$ and $\Sigma(\mathbf{m};\mathbf{q})$ be two equidimensional generalized pseudoellipsoids, $\mathbf{n},\mathbf{m}\in\mathbb{N}^{l}$, $\mathbf{p},\mathbf{q}\in\mathbb{(R_{+})}^{l}$ (where $p_{k},\;q_{k}\neq 1\;for\;2\leq k\leq l$). Let $h:\Sigma(\mathbf{n};\mathbf{p})\rightarrow\Sigma(\mathbf{m};\mathbf{q})$ be a biholomorphic linear isomorphism between $\Sigma(\mathbf{n};\mathbf{p})$ and $\Sigma(\mathbf{m};\mathbf{q})$. Then there exists a permutation $\sigma\in S_{r}$ such that $n_{\sigma(i)}=m_{i},\;p_{\sigma(i)}=q_{i}$ and

[TABLE]

where $U_{i}$ is a unitary transformation of $\mathbb{C}^{m_{i}}(m_{i}=n_{\sigma(i)})$ for $1\leq i\leq r$.

Define

[TABLE]

[TABLE]

Then we have the following lemma.

Lemma 3.2**.**

Suppose $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ are two equidimensional generalized Fock-Bargmann-Hartogs domains, $f\;:D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ is a biholomorphic mapping. Then we have $f(V_{1})\subseteq V_{2}$ and $f|_{V_{1}}\;:V_{1}\rightarrow V_{2}$ is biholomorphic. Consequently $n_{0}=m_{0}$.

Proof.

Let $f(z,0)=(f_{1}(z),f_{2}(z))$, then we get $\sum\limits_{i=1}^{l}{\|f_{2i}\|}^{2q_{i}}<e^{-\nu{\left\lVert f_{1}(z)\right\rVert}^{2}}\leq 1.$ Then we obtain that the bounded entire mapping $f_{2i}(z)$ on $\mathbb{C}^{n_{0}}$ is constant $(1\leq i\leq l)$ by Liouville’s Theorem. Since $f(z)$ is biholomorphic, $f_{1}(z)$ is an unbounded function. Hence there exist $\{z_{k}\}$ such that $f_{1}(z_{k})\rightarrow\infty$ as $k\rightarrow\infty$. It implies $f_{2}(z)\equiv 0$. This proves $f(V_{1})\subseteq V_{2}$. Similarly, by making the same argument for $f^{-1}$, we have $f^{-1}(V_{2})\subseteq V_{1}$. Namely, $f|_{V_{1}}\;:V_{1}\rightarrow V_{2}$ is biholomorphic. Hence $n_{0}=m_{0}$. ∎

Now we give the proof of Theorem 1.4.

The proof of Theorem 1.4.

Let $f(0,0)=(a,b)$ (thus $b=0$ by Lemma 3.2) and define

[TABLE]

Obviously, $\phi\in\mathrm{Aut}(D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu))$ and $\phi\circ f(0,0)=(0,0)$. Then $\phi\circ f$ is linear by Theorem 2.10. We describe $\phi\circ f$ as follows:

[TABLE]

According to Lemma 3.2, we have $f(z,0)=(f_{1}(z),0)$. Thus $B=0$. Since $g:=\phi\circ f$ is biholomorphic, $A$ and $D$ are invertible matrices. We write $g(z,w)$ as follows:

[TABLE]

which implies that

[TABLE]

Set $\Sigma(\mathbf{n};\mathbf{p})=\{(w_{(1)},\cdots,w_{(l)})\in\mathbb{C}^{n_{1}}\times\cdots\times\mathbb{C}^{n_{l}}:\sum\limits_{j=1}^{l}{\|w_{(j)}\|}^{2p_{j}}<1\}.$ Then, if $\sum\limits_{j=1}^{l}{\|w_{(j)}\|}^{2p_{j}}<e^{-\mu{\left\lVert 0\right\rVert}^{2}}=1$, we obtain

[TABLE]

and if $\sum\limits_{j=1}^{l}{\|w_{(j)}\|}^{2q_{j}}<e^{-\nu{\left\lVert 0\right\rVert}^{2}}=1$, we have

[TABLE]

Therefore, we conclude that the mapping $g_{2}(w):\Sigma(\mathbf{n};\mathbf{p})\rightarrow\Sigma(\mathbf{m};\mathbf{q})$ given by

[TABLE]

is a biholomorphic linear mapping. By Lemma 3.1, $g_{2}$ can be expressed in the form:

[TABLE]

where $\sigma\in S_{l}$ is a permutation with $n_{\sigma(j)}=m_{j}$, $p_{\sigma(j)}=q_{j}\;(j=1,\cdots,l)$ and $\Gamma_{i}\in\mathcal{U}(m_{i})\;(1\leq i\leq l)$. Hence $g$ can be rewritten as follows:

[TABLE]

Next we prove that $C=0.$ The linearity of $g$ yields that $g(bD_{n_{0}}^{\mathbf{n},\mathbf{p}})=bD_{m_{0}}^{\mathbf{m},\mathbf{q}}.$ Let $(0,w)=(0,0,\cdots,w_{(j)},0,\cdots,0)\in bD_{n_{0}}^{\mathbf{n},\mathbf{p}}$, namely, ${\|w_{(j)}\|}^{2}=(e^{-\mu{\left\lVert 0\right\rVert}^{2}})^{\frac{1}{p_{j}}}=1$. As $\Gamma_{j}$ $(1\leq j\leq l)$ are unitary matrices, moreover, assuming $\sigma(i_{0})=j$, we have

[TABLE]

This implies $w_{(j)}C_{j1}=0$ for all ${\|w_{(j)}\|}^{2}=1$. So $C_{j1}=0\;(1\leq j\leq l).$ Thus we have

[TABLE]

Lastly, we show $\sqrt{\frac{\nu}{\mu}}A\in\mathcal{U}(n)$ $(n:=n_{0}=m_{0})$. For $z\in\mathbb{C}^{n_{0}}$, take $(w_{(1)},\cdots,w_{(l)})$ such that $e^{-\mu{\left\lVert z\right\rVert}^{2}}=\sum\limits_{j=1}^{l}{\|w_{(j)}\|}^{2p_{j}}.$ By $g(bD_{n_{0}}^{\mathbf{n},\mathbf{p}})=bD_{m_{0}}^{\mathbf{m},\mathbf{q}}$, we have $\sum\limits_{j=1}^{l}{\|w_{(\sigma(j))}\Gamma_{j}\|}^{2q_{j}}=e^{-\mu{\left\lVert zA\right\rVert}^{2}}.$ Since $\Gamma_{j}(j=1,\cdots,l)$ are unitary matrices, we get

[TABLE]

Therefore, $\nu\|zA\|^{2}=\mu{\left\lVert z\right\rVert}^{2}$ $(z\in\mathbb{C}^{n})$. Then we get $\sqrt{\frac{\nu}{\mu}}A\in\mathcal{U}(n)$, and the proof is completed. ∎

The proof of Corollary 1.5.

In fact, the significance of the above $\phi$ is just to ensure $\phi\circ f(0)=0$. Then the proof of Theorem 1.4 implies that Corollary 1.5 is obvious. ∎

The proof of the Theorem 1.6.

Obviously, $\varphi_{A},\varphi_{D}\textrm{ and }\varphi_{a}$ are biholomorphic self-mappings of $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. On the other hand, for $\varphi\in\mathrm{Aut}(D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu))$, we assume $\varphi(0,0)=(a,b)$ (then $b=0$ by Lemma 3.2). Hence $\varphi_{-a}\circ\varphi$ preserves the origin. Then by Corollary 1.5, we obtain $\varphi_{-a}\circ\varphi=\varphi_{D}\circ\varphi_{A}$ for some $\varphi_{A},\varphi_{D}$. Hence $\varphi=\varphi_{a}\circ\varphi_{D}\circ\varphi_{A}$, and the proof is complete. ∎

The proof of Theorem 1.7.

Let $f$ be a proper holomorphic mapping between two equidimensional generalized Fock-Bargmann-Hartogs domains $D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ and $D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$. Then by Th. 2.7, $f$ extends holomorphically to a neighborhood $\Omega$ of $\overline{D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)}$ with

[TABLE]

Then by Proposition 3.1 and Lemma 1.3 in Pinčuk [19], we have

[TABLE]

where M:=\big{\{}z\in\Omega,\mathrm{det}(\frac{\partial f_{i}}{\partial z_{j}})=0\big{\}} is the zero locus of the complex Jacobian of the holomorphic mapping $f$ on $\Omega$.

If $M\cap bD_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\neq\emptyset$, then, from min $\{n_{1+\epsilon},n_{2},\cdots,n_{l}\}\geq 2$, we have $M\cap b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\neq\emptyset$. Take an irreducible component $M^{\prime}$ of $M$ with $M^{\prime}\cap b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\neq\emptyset$. Then the intersection $E_{M^{\prime}}$ of $M^{\prime}$ with $b_{0}D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$ is a real analytic submanifold of dimensional $2(n_{0}+n_{1}+\cdots+n_{l})-3$ on a dense, open subset of $E_{M^{\prime}}$. By (3.7), we have $f(E_{M^{\prime}})\subset b_{1}D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)\cup b_{2}D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$. Hence

[TABLE]

where $Pr_{i}(D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)):=\{(z,w_{(1)},\cdots,w_{(l)})\in D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu),\;\|w_{(i)}\|=0\}$ ($1+\delta\leq i\leq l$), by the uniqueness theorem. Since $\mathrm{codim}M^{\prime}=1$, $\mathrm{codim}[\bigcup\limits_{j=1+\delta}^{l}Pr_{i}(D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu))]\geq\textrm{min}\{m_{1+\delta},\cdots,m_{l},m_{1}+\cdots+m_{l}\}\geq 2$ and $f:D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ is proper, this is contradiction with (3.8). Thus we have $M\cap bD_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)=\emptyset$.

Let $S:=M\cap D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)$. Hence we have

[TABLE]

If $S\neq\emptyset$, then $S$ is a complex analytic set in $\mathbb{C}^{n_{0}+n_{1}+\cdots+n_{l}}$ also. For any $(z,w)\in S$, we have ${|w_{ln_{l}}}|^{2p_{l}}\leq\sum\limits_{j=1}^{l}{\|w_{(j)}\|}^{2p_{j}}\leq e^{-\mu{\left\lVert z\right\rVert}^{2}}\leq 1.$ Thus

[TABLE]

where $w=(w^{\prime},w_{ln_{l}})$. Then $S$ is an algebraic set of $\mathbb{C}^{n_{0}+n_{1}+\cdots+n_{l}}$ by §7.4 Th. 3 of Chirka [5].

Suppose $S_{1}$ is an irreducible component of $S$. Let $\overline{S_{1}}$ be the closure of $S_{1}$ in $\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}}$. Then by §7.2 Prop. 2 of Chirka [5], $\overline{S_{1}}$ is a projective algebraic set and $\dim\overline{S_{1}}=n_{0}+n_{1}+\cdots+n_{l}-1$. Let $[\xi,z,w]$ be the homogeneous coordinate in $\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}}$, we embed $\mathbb{C}^{n_{0}+n_{1}+\cdots+n_{l}}$ into $\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}}$ as the affine piece $U_{0}=\{[\xi,z,w]\in\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}},\;\xi\neq 0\}$ by $(z,w)\hookrightarrow[1,z,w]$. Then we have

[TABLE]

Let $H=\{\xi=0\}\subset\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}}$. Consider another affine piece $U_{1}=\{[\xi,z,w]\in\mathbb{P}^{n_{0}+n_{1}+\cdots+n_{l}},\;z_{1}\neq 0\}$ with affine coordinate $(\zeta,t,s)=(\zeta,t_{2},\cdots,t_{n_{0}},s_{(1)},\cdots,s_{(l)})$. Let $t^{\prime}=(1,t_{2},\cdots,t_{n_{0}})$. Since $\frac{{\|w_{(j)}\|}^{2p_{j}}}{{|\xi|}^{2p_{j}}}=\frac{{\|w_{(j)}\|}^{2p_{j}}}{{|z_{1}|}^{2p_{j}}}\frac{{|z_{1}|}^{2p_{j}}}{{|\xi|}^{2p_{j}}}=\frac{{\|s_{(j)}\|}^{2p_{j}}}{{|\zeta|}^{2p_{j}}}$ and $e^{-\mu\frac{{\|z\|}^{2}}{{|\xi|}^{2}}}=e^{-\mu\frac{{\|z\|}^{2}}{{|z_{1}|}^{2}}\frac{{|z_{1}|}^{2}}{{|\xi|}^{2}}}=e^{-\mu\frac{1+{|t_{2}|}^{2}+\cdots+{|t_{n_{0}}|}^{2}}{{|\zeta|}^{2}}}$, we obtain

[TABLE]

Let $S^{\prime}=\overline{S_{1}}\cap U_{1}$ and $H_{1}=H\cap U_{1}=\{\zeta=0\}$ (note $\xi=\frac{\zeta}{z_{1}}$). For every $u\in S^{\prime}\cap H_{1}$, there exists a sequence of points $\{u_{k}\}\subset\overline{S_{1}}\cap((U_{0}\cap U_{1})\backslash H_{1})$ such that $u_{k}\rightarrow u\;(k\rightarrow\infty)$, The formula (3.10) implies

[TABLE]

Since $u\in H_{1}$, that means $\zeta(u)=0$ and $\zeta{(u_{k})}\rightarrow 0\;(k\rightarrow\infty)$. Therefore we have ${\|s_{(j)}(u)\|}^{2p_{j}}\leq 0\;(1\leq j\leq l)$ as $k\rightarrow\infty$. Hence

[TABLE]

Then $\dim(S^{\prime}\cap H_{1})\leq n_{0}-1$. Shafarevich [20] §6.2 Th. 6 implies

[TABLE]

This means $\dim S^{\prime}\leq n_{0}$, and thus $n_{0}+n_{1}+\cdots+n_{l}-1=\dim S^{\prime}\leq n_{0}$. Therefore, we get $n_{1}+\cdots+n_{l}\leq 1$, a contradiction with assumption min $\{n_{1+\epsilon},n_{2},\cdots,n_{l},n_{1}+\cdots+n_{l}\}\geq 2$.

Therefore, $S=\emptyset$ and thus $f$ is unbranched. Since the generalized Fock-Bargmann-Hartogs domain is simply connected, $f:D_{n_{0}}^{\mathbf{n},\mathbf{p}}(\mu)\rightarrow D_{m_{0}}^{\mathbf{m},\mathbf{q}}(\nu)$ is a biholomorphism. The proof is completed. ∎

Acknowledgments We sincerely thank the referees, who read the paper very carefully and gave many useful suggestions. E. Bi was supported by the Natural Science Foundation of Shandong Province, China (No.ZR2018BA015), and Z. Tu was supported by the National Natural Science Foundation of China (No.11671306).